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1、定义
基于动量矩定理，用高速回转体的动量矩敏感壳体相对惯性空间绕正交于自转轴的一个或二个轴的角运动检测装置
2、特性
定轴性、进动性

定轴性：转动的陀螺，如未受到外力矩的作用，则其主轴的方向永不改变
进动性：当陀螺在垂直于..." />
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1、定义
基于动量矩定理，用高速回转体的动量矩敏感壳体相对惯性空间绕正交于自转轴的一个或二个轴的角运动检测装置
2、特性
定轴性、进动性

定轴性：转动的陀螺，如未受到外力矩的作用，则其主轴的方向永不改变
进动性：当陀螺在垂直于...">
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1、定义
基于动量矩定理，用高速回转体的动量矩敏感壳体相对惯性空间绕正交于自转轴的一个或二个轴的角运动检测装置
2、特性
定轴性、进动性

定轴性：转动的陀螺，如未受到外力矩的作用，则其主轴的方向永不改变
进动性：当陀螺在垂直于...">
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                                <h2>
                                    5.2 GNC 硬件——陀螺仪
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                                    2024-05-10, 1661 words, 6 min read
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                                        <h2 id="一-陀螺仪">一、陀螺仪</h2>
<h4 id="1-定义">1、定义</h4>
<p>基于<strong>动量矩定理</strong>，用高速回转体的动量矩敏感壳体相对惯性空间绕正交于自转轴的一个或二个轴的<strong>角运动检测装置</strong></p>
<h4 id="2-特性">2、特性</h4>
<p>定轴性、进动性</p>
<ul>
<li>定轴性：转动的陀螺，如未受到外力矩的作用，则其主轴的方向永不改变</li>
<li>进动性：当陀螺在垂直于主轴的方向受到外力矩的作用时，陀螺的主轴则绕第三个正交轴转动，这种运动特性称为陀螺的进动性，也叫受迫进动。</li>
</ul>
<h4 id="3-进动角速度">3、进动角速度</h4>
<p>1、进动轴：将陀螺仪进动所绕的轴叫做进动轴</p>
<p>2、进动角速度方向：转子转动的动量矩<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>H</mi><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{H}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20533em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em;">H</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span>和外力矩<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mover accent="true"><mi>M</mi><mo stretchy="true">→</mo></mover></mrow><annotation encoding="application/x-tex">\overrightarrow{M}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.20533em;vertical-align:0em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.20533em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span></span></span><span class="svg-align" style="top:-3.6833299999999998em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="height:0.522em;min-width:0.888em;"><svg width='400em' height='0.522em' viewBox='0 0 400000 522' preserveAspectRatio='xMaxYMin slice'><path d='M0 241v40h399891c-47.3 35.3-84 78-110 128
-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
 151.7 139 205zm0 0v40h399900v-40z'/></svg></span></span></span></span></span></span></span></span></span>的叉乘方向</p>
<img src="http://cos.pansis.site/202310161526622.png/abc123" alt="image-20231016152631546" style="zoom:33%;" />
<p>3、进动角速度大小：</p>
<img src="http://cos.pansis.site/202310161526411.png/abc123" alt="image-20231016152659368" style="zoom:33%;" />
<h2 id="二-陀螺仪分类">二、陀螺仪分类</h2>
<h4 id="1-按功能分类">1、按功能分类</h4>
<ul>
<li>位置陀螺仪：能够测量飞行器角度信息</li>
<li>速率陀螺仪：能够测量飞行器角速度信息</li>
<li>积分陀螺仪：能够测量飞行器角度和角速度信息</li>
</ul>
<h4 id="2-按转子轴相对于底座的转动自由度分类">2、按转子轴相对于底座的转动自由度分类</h4>
<p>1、陀螺仪自由度数目：自转轴可绕其自由旋转的正交轴数目</p>
<p>2、根据自由度划分分为：二自由度陀螺仪、单自由度陀螺仪</p>
<p>3、结构名称</p>
<p>二自由度陀螺仪</p>
<img src="http://cos.pansis.site/202310130756302.png/abc123" alt="image-20231013075649046" style="zoom:23%;" />
<p>单自由度陀螺仪</p>
<img src="http://cos.pansis.site/202310130757382.png/abc123" alt="image-20231013075714127" style="zoom:20%;" />
<h4 id="3-按产生陀螺效应的原理分类">3、按产生陀螺效应的原理分类</h4>
<p>转子陀螺仪、激光陀螺仪、光纤陀螺仪、MEMS陀螺仪</p>
<h4 id="4-按对陀螺转子支撑方式不同分类">4、按对陀螺转子支撑方式不同分类</h4>
<p>滚珠轴承陀螺仪、气浮陀螺仪、液浮陀螺仪、弹性支撑陀螺仪、静电支撑陀螺仪</p>
<ul>
<li>所有支撑的目标：消除支撑轴上的摩擦力矩</li>
<li>气浮陀螺仪：分为静压气浮支撑和动压气浮支撑</li>
<li>液浮陀螺仪：分为静浮力支撑和静压力支撑</li>
<li>弹性支撑陀螺仪：分为弹性支撑和挠性支撑</li>
<li>静电支撑陀螺仪：用静电力支承高速旋转的转子</li>
</ul>
<h2 id="三-位置陀螺仪">三、位置陀螺仪</h2>
<h4 id="1-分类">1、分类</h4>
<ul>
<li>垂直陀螺仪：用来测量飞行器的滚转角和俯仰角</li>
<li>方向陀螺仪：用来测量飞行器的偏航角和俯仰角</li>
</ul>
<h4 id="2-垂直陀螺仪">2、垂直陀螺仪</h4>
<p>1、各轴方向</p>
<p>转子轴与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">Oy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span></span> 轴重合，内环轴与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Ox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault">x</span></span></span></span> 轴重合，外环轴与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">Oz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span></span></span></span> 轴重合</p>
<p>2、测量角</p>
<img src="http://cos.pansis.site/202405081843062.png/abc123" alt="image-20240508184247866" style="zoom:30%;" />
<ul>
<li>滚转角
<ul>
<li>输出电位器位于内环轴上。</li>
<li>滚转角 𝛾，电位器的输出电压 <span class='katex-error' title='Error: Font metrics not found for font: .'>𝑢_𝛾</span> 的关系：<span class='katex-error' title='Error: Font metrics not found for font: .'>𝛾=k_𝛾𝑢_𝛾</span></li>
</ul>
</li>
<li>俯仰角
<ul>
<li>输出电位器位于外环轴上。</li>
<li>俯仰角 𝜃，电位器的输出电压 <span class='katex-error' title='Error: Font metrics not found for font: .'>𝑢_𝜃</span> 的关系：<span class='katex-error' title='Error: Font metrics not found for font: .'>𝜃=k_𝜃𝑢_𝜃</span></li>
</ul>
</li>
</ul>
<h4 id="3-方向陀螺仪">3、方向陀螺仪</h4>
<p>1、各轴方向</p>
<p>转子轴与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Ox</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault">x</span></span></span></span> 轴重合，内环轴与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mi>z</mi></mrow><annotation encoding="application/x-tex">Oz</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span></span></span></span> 轴重合，外环轴与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">Oy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span></span> 轴重合</p>
<p>2、测量角</p>
<img src="http://cos.pansis.site/202405081844401.png/abc123" alt="image-20240508184428321" style="zoom:40%;" />
<ul>
<li>偏航角
<ul>
<li>输出电位器位于外环轴上。</li>
<li>偏航角 𝜓 ，电位器的输出电压 <span class='katex-error' title='Error: Font metrics not found for font: .'>𝑢_𝜓</span> 的关系：<span class='katex-error' title='Error: Font metrics not found for font: .'>𝜓=k_𝜓𝑢_𝜓</span></li>
</ul>
</li>
<li>俯仰角
<ul>
<li>输出电位器位于内环轴上。</li>
<li>俯仰角 𝜃，电位器的输出电压 <span class='katex-error' title='Error: Font metrics not found for font: .'>𝑢_𝜃</span> 的关系：<span class='katex-error' title='Error: Font metrics not found for font: .'>𝜃=k_𝜃𝑢_𝜃</span></li>
</ul>
</li>
</ul>
<h4 id="4-陀螺仪的漂移">4、陀螺仪的漂移</h4>
<p>1、漂移定义：在工作过程中，始终会有一定量 值的干扰力矩作用在陀螺仪上，从而产生进动，使自转轴在惯性空间中逐渐偏离原来的方位</p>
<p>2、漂移角速度</p>
<img src="http://cos.pansis.site/202310161540390.png/abc123" alt="image-20231016154010322" style="zoom:33%;" />
<p>3、陀螺漂移的快慢用单位时间内的漂移角度，即用漂移角速度来表示</p>
<p>4、干扰力矩的分类</p>
<p>（1）与加速度无关的常值力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">M_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><br>
（2）与加速度一次方成比例的力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><br>
（3）与加速度平方成比例的力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>
<p>常值力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">M_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>主要来源
<ul>
<li>输电装置（如软导线）的扭力矩</li>
<li>电磁元件（如传感器和力矩器）的电磁干扰力矩</li>
<li>空气轴承的常值涡流力矩等</li>
</ul>
</li>
</ul>
</li>
<li>
<p>加速度一次方成比例的力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>
<p>产生原因：陀螺仪<strong>质心与转轴存在偏差</strong>，当飞行器加速运动时，产生的<strong>惯性力</strong>导致干扰力矩的出现</p>
</li>
<li>
<p>干扰力矩<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>公式：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub><mo>=</mo><mi>D</mi><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">M_1=D×F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span></p>
<p>​	其中，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">D</span></span></span></span>为陀螺仪质心静不平衡位移矢量，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span> 为惯性力</p>
</li>
</ul>
</li>
<li>
<p>加速度平方成比例的力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>
<p>产生原因：陀螺仪非等刚度在两个正交加速度作用下产生</p>
</li>
<li>
<p>干扰力矩<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>公式：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub><mo>=</mo><msub><mi>L</mi><mi>s</mi></msub><mo>×</mo><mi>F</mi><mo>=</mo><mo>(</mo><mi>C</mi><mo separator="true">⋅</mo><mi>F</mi><mo>)</mo><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">M_2=L_s×F=(C·F)×F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span></p>
<p>​		其中， <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span></span></span></span> 为 3×3 的柔性系数张量</p>
</li>
</ul>
</li>
<li>
<p>总干扰力矩 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mi>d</mi></msub></mrow><annotation encoding="application/x-tex">M_d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<ul>
<li>公式：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mi>d</mi></msub><mo>=</mo><msub><mi>M</mi><mn>0</mn></msub><mo>+</mo><msub><mi>M</mi><mn>1</mn></msub><mo>+</mo><msub><mi>M</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_d=M_0+M_1+M_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10903em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
</ul>
</li>
</ul>
<p>​		（以转子轴为z轴的陀螺仪为例，只需要x轴和y轴的干扰力矩项）</p>
<figure data-type="image" tabindex="1"><img src="http://cos.pansis.site/202405081916355.png/abc123" alt="image-20240508191646290" loading="lazy"></figure>
<p>5、漂移角速率</p>
<p>（以转子轴为z轴的陀螺仪为例，H沿z轴正方向）</p>
<figure data-type="image" tabindex="2"><img src="http://cos.pansis.site/202405081921512.png/abc123" alt="image-20240508192122432" loading="lazy"></figure>
<p>其中随机漂移角速度<span class='katex-error' title='Error: Font metrics not found for font: .'>𝜀_𝑥</span>、<span class='katex-error' title='Error: Font metrics not found for font: .'>𝜀_𝑦</span> 表示不可建模的漂移角速度</p>
<h2 id="四-速率陀螺仪">四、速率陀螺仪</h2>
<p>利用陀螺的进动性，单自由度陀螺仪可做成速率陀螺仪</p>
<h4 id="1-分类-2">1、分类</h4>
<p>弹簧式和液浮式两种速率陀螺仪</p>
<h4 id="2-原理">2、原理</h4>
<img src="http://cos.pansis.site/202405081927089.png/abc123" alt="image-20240508192729022" style="zoom:33%;" />
<p>飞行器以角速度 <span class='katex-error' title='Error: Font metrics not found for font: .'>𝜔_y</span> 转动时，迫使转子以相同的角速度进动，产生陀螺力矩𝑀′</p>
<p>框架在𝑀𝑀′的作用下绕 𝑂𝑥 轴转动，当框轴转过角度𝛽增大到使弹簧的反抗力矩与陀螺力矩相互平衡时，框轴停止转动。角度 𝛽 由电位器测量得到。</p>
<h4 id="3-速率陀螺仪的动力学方程和传递函数">3、速率陀螺仪的动力学方程和传递函数</h4>
<p>1、动力学方程</p>
<img src="http://cos.pansis.site/202405081929876.png/abc123" alt="image-20240508192929809" style="zoom:40%;" />
<p>2、传递函数</p>
<img src="http://cos.pansis.site/202405081929981.png/abc123" alt="image-20240508192958922" style="zoom:40%;" />
<p>3、稳态下，速率陀螺仪电位器测量角速度与测量角速率的关系</p>
<img src="http://cos.pansis.site/202405081930413.png/abc123" alt="image-20240508193033373" style="zoom:47%;" />
<h4 id="4-考虑弹性振动">4、考虑弹性振动</h4>
<ul>
<li>在考虑<strong>高细长比</strong>火箭的<strong>弹性振动</strong>情况下，速率陀螺仪的敏感信号包括<strong>等效刚体纵轴转动</strong>和弹<br>
性变形带来的弹体纵轴的<strong>附加转动</strong></li>
<li>解决方案：在多个不同位置安装速率陀螺，能够改变多次振型的等效斜率，从而满足弹性振动稳定的要求</li>
</ul>
<h2 id="五-陀螺仪在制导控制系统中的作用">五、陀螺仪在制导控制系统中的作用</h2>
<h4 id="1-飞行器制导回路">1、飞行器制导回路</h4>
<img src="http://cos.pansis.site/202405081933611.png/abc123" alt="image-20240508193356550" style="zoom:50%;" />
<p>陀螺仪主要参与飞行器控制系统的校正</p>
<h4 id="2-回路">2、回路</h4>
<img src="http://cos.pansis.site/202405081938095.png/abc123" alt="image-20240508193827867" style="zoom:50%;" />
<img src="http://cos.pansis.site/202405081938569.png/abc123" alt="image-20240508193856430" style="zoom:40%;" />
<h4 id="3-简化回路">3、简化回路</h4>
<p>忽略执行机构的特性，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>W</mi><mi>g</mi></msub><mo>(</mo><mi>s</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">W_g(s)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mclose">)</span></span></span></span>只包含放大环节（其时间常数𝑇比飞行器时间常数𝑇__𝑑小很多）</p>
<img src="http://cos.pansis.site/202405081941580.png/abc123" alt="image-20240508194128389" style="zoom:50%;" />
<p>闭环系统传递函数</p>
<p>通过选择速率陀螺仪（包含校正网络）的传递系数<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>𝐾</mi><mi>𝑔</mi></msub></mrow><annotation encoding="application/x-tex">𝐾_𝑔</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>，使参数满足要求</p>
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